Register to reply

Is Euclidean space an affine space?

by BruceW
Tags: affine, euclidean, space
Share this thread:
Oct2-13, 05:27 AM
HW Helper
BruceW's Avatar
P: 3,447
Hi everyone,

I have a question that I'm not sure about. I wanted to know if it is standard to think of Euclidean space as a linear vector space, or a (more general) affine space? In some places, I see Euclidean space referred to as an affine space, meaning that the mathematical definition of the space allows us to make translations without affecting our system.

But on the other hand, if we have a Lagrangian like ##1/2 \ m v^2+mgy## then a translation ##y \rightarrow y+d## changes the Lagrangian to ##L \rightarrow L+mgd##. Now, I know that this does not affect the physics of the system, since it does not matter if we add a constant to the Lagrangian. But when people talk about Noether's theorem, they say that due to the form of this Lagrangian, we do not have 'translational invariance'. So are they including the Lagrangian as a physical observable of the system?? And so, is our space a linear vector space, not an affine space? Maybe they are using the term 'translational invariance' to mean something different to the 'translational invariance' that allows us to call our space affine? (and if so, then that is pretty darn confusing).

thanks in advance :)
Phys.Org News Partner Physics news on
Detecting neutrinos, physicists look into the heart of the Sun
Measurement at Big Bang conditions confirms lithium problem
Researchers study gallium to design adjustable electronic components

Register to reply

Related Discussions
Topological space, Euclidean space, and metric space: what are the difference? Calculus & Beyond Homework 9
Relations of an affine space with R^n , and the construction of Euclidean space Differential Geometry 10
Affine n-space Linear & Abstract Algebra 6
Euclidean space, euclidean topology and coordinate transformation Differential Geometry 8
Metric space and subsets of Euclidean space Calculus & Beyond Homework 18